Question

1. True of False: Write "T" in the blank if the statement is always true. ___ If f(x) is a continuous function on closed interval [a, b] , then it must have both an absolute maximum and an absolute minimum on [a, b]. _ If f '(3) = 0, then f(3) must be a local and /or absolute extreme value. _ Critical numbers are only where the derivative is zero. _ If f "(x) > 0 on (-?, ?), then f(x) is always concave up. _____ If f '(x) = g'(x) for all real numbers, then f(x) = g(x). 2. Draw a sketch of a function with domain all reals that is continuous everywhere and fits the properties: f(-1) = 0, f(5) = 0, f '(x) > 0 for x < 4, f '(x) < 0 for x > 4 3. Verify that f(x) = 5x - x³ satisfies the hypothesis of the Mean Value Theorem on the interval [-2, 3] and find all values of c in the interval that satisfies the conclusion of the theorem (exact value). Show your analytic work.

          1. True of False: Write "T" in the blank if the statement is always true.
_______ If f(x) is a continuous function on closed interval [a, b] , then it must have both an absolute maximum and an absolute minimum on [a, b].
_______ If f '(3) = 0, then f(3) must be a local and /or absolute extreme value.
_______ Critical numbers are only where the derivative is zero.
_______ If f "(x) > 0 on (-?, ?), then f(x) is always concave up.
_______ If f '(x) = g'(x) for all real numbers, then f(x) = g(x).
2. Draw a sketch of a function with domain all reals that is continuous everywhere and fits the properties:
f(-1) = 0, f(5) = 0, f '(x) > 0 for x < 4, f '(x) < 0 for x > 4
3. Verify that f(x) = 5x - x³ satisfies the hypothesis of the Mean Value Theorem on the interval [-2, 3] and find all values of c in the interval that satisfies the conclusion of the theorem (exact value). Show your analytic work.

1. True of False: Write "T" in the blank if the statement is always true.
If f(x) is a continuous function on closed interval [a, b] , then it must have both an absolute maximum and an absolute minimum on [a, b].
If f '(3) = 0, then f(3) must be a local and /or absolute extreme value.
Critical numbers are only where the derivative is zero.
If f "(x) > 0 on (-?, ?), then f(x) is always concave up.
If f '(x) = g'(x) for all real numbers, then f(x) = g(x).
2. Draw a sketch of a function with domain all reals that is continuous everywhere and fits the properties:
f(-1) = 0, f(5) = 0, f '(x) > 0 for x < 4, f '(x) < 0 for x > 4
3. Verify that f(x) = 5x - x³ satisfies the hypothesis of the Mean Value Theorem on the interval [-2, 3] and find all values of c in the interval that satisfies the conclusion of the theorem (exact value). Show your analytic work.

Added by Nicole A.

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